Recently I had a few papers published, and I suppose in response to this, I received a request from Zentralblatt to be a reviewer. They ask some general questions about what I would feel comfortable reviewing, and if I would be willing to receive papers electronically. My concern about just signing up stems from the fact that I am only a second year graduate student. So my question is

I wrote my first Math Review (MathSciNet) review when I was a grad student. Here's my advice: do it, but don't spend too much time on it. Remember that some reviews are of the form "copy out the first para of the article", and it would be good if you did better than that, but definitely don't spend 2 weeks worrying about a review before writing it, and then another 2 weeks writing it!
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Kevin BuzzardFeb 8 '10 at 7:46

I ever received such kind of request to review some papers in the field of computational mathematics and algorithm.. I just disregard all of them
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Shizhuo ZhangFeb 8 '10 at 8:01

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Well, Kevin this depends on the importance of the article and standing of the journal. If one gets a paper claiming a significant progress toward the Riemann conjecture one absolutely has a responsibility to spend more than two weeks reviewing it. The present carelessness of referees and of reviewers is so big a problem that being mainstream means being irresponsible. I think we should pressure on journals and reveiwing services to be more considerate in allowing additional time, when needed. I had pretty bad experience with services in few cases in which I was slightly late.
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Zoran SkodaFeb 10 '11 at 14:14

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@Zoran: What you are saying is certainly true for peer review, but I don't see how it applies to review databases of published articles like Zentralblatt or Math Reviews. The paper is already supposed to be vetted for correctness by the journal where it was accepted and published. It's not the job of a Zentralblatt reviewer to do that again, they are only supposed to provide an accurate and informative summary of the content of the paper.
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Emil JeřábekFeb 11 '11 at 11:44

4 Answers
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Reviewing is completely voluntary in all senses. I don't know the percentage of research active mathematicians who have ever reviewed for MathReviews or Zentralblatt, but it is certainly bounded away from 100%. It is relatively rare for a research mathematician to write reviews in any quantity over a period of more than a few years: I think it's something that many of us try out for a while but don't stick with in the long run. For a personal touch, I am a 2003 PhD, I have written about five math reviews, and I have gotten overdue notices on about four more. I would rather write nice, insightful reviews for half of the papers that I get sent, but I haven't yet figured out what do about that.

I agree with Tim Porter that, as a graduate student, you should discuss this with your advisor. I think that if any of my students asked me about it, I would mildly discourage them from writing reviews, for the following reasons:

(i) It is not closely enough related to your thesis work to be a good use of a mid-to-late career grad student's time.

(ii) Prospective employers are generally not going to be more excited about your application because of the reviews you have written.

(iii) Most graduate students -- even ones who have already done some research work of their own -- don't have a broad enough perspective to write insightful reviews. (Or, more positively put, they will have a broader perspective later on and will probably write only a bounded number of reviews in their life no matter when they start.)

Finally, I should say that refereeing papers is a totally different story. I feel that it is an ethical (and karmic) responsibility to referee at least as many papers as one submits. A graduate student can make a good referee for some papers, because they are less likely to feel that it is beneath them to read the paper line by line and really make sure it is correct.

I would agree with this, but with a caveat. Firstly, I enjoy reviewing; perhaps I've been lucky that maybe 75% of what I get sent I would (or already have) read anyway to keep on top of the literature. So it's not too much hassle to write a few paragraphs once every other month. I would strongly encourage people to review (and to write good reviews); some comments here have been a bit negative! But, it does take time, so, yes, I think I wouldn't be in any rush to start!
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Matthew DawsFeb 8 '10 at 10:52

I actually like reviewing, and I say do it!
Reviewing forces you to read papers carefully, and to broaden your view of your field. This seems especially important when one is a grad student. I am perhaps atypical as a reviewer, in that I spend huge amounts of time reviewing papers, because it's exciting for me to see all the great work other people are doing! I know it makes me, if not a better mathematician, then at least a more widely read one. So I write maybe a few dozen reviews for Zentralblatt each year. The nasty part is when you get sent papers to review which are rubbish, in which case (I think) you do a community service by pointing out why that is so.

Actually, all reviews are written by research mathematicians, since MR/Zentralblatt typically only invite those who have published papers.
In fact, it is not a great burden. As everyone knows, the best way to understand some mathematics is to explain it others. If you get a paper to review that you have read, or were planning to read, then writing a review doesn't take long. If you get a paper that you weren't planning to read, then reviewing it broadens your understanding.
So my advice is to begin reviewing when invited, although no harm will be done if you prefer to wait until you are more experienced.
Since all of us benefit from reviews, all of us have an ethical obligation to contribute for at least part of our careers.

@JSM: Although I do write for MathReviews currently (in fact I just finished one tonight), I do not feel ethically obligated to do so. I do not want to try to talk you out of it, but I feel that on a purely philosophical level there must be some fallacy here: are we really ethically obligated to do everything that would benefit the community? It seems hard to believe.
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Pete L. ClarkFeb 9 '10 at 6:49

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No, you are not obligated to do everything that would benefit the community, but that's not the point here. When you benefit from the volunteer efforts of others, you have an obligation to contribute your share. From another (Kantian) point of view, if everyone decided not to review there would be no reviews, which would be bad for everyone. I don't see much of a difference between reviewing and refereeing in this respect.
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JS MilneFeb 10 '10 at 12:38

Some people are in better situations to write reviews so they write a disproportionate amount. If you ever get a chance to read the Steenrod compilation "Reviews of papers in Algebraic and Differential Topoogy, Topological Groups, and Homological Algebra" (two big fat blue AMS books) do. They're a wonderful example, especially if you're trying to get a sense for the history of the subject.
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Ryan BudneyFeb 10 '10 at 19:24

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Moreover, delays in refereeing can have dire consequences -- they can result in excellent mathematicians not finding their next job. Reviewing delays are at worst mildly inconvenient. My one paper on Shimura curves appeared in 2009 and hasn't been reviewed. I can wait. Anyone else who is interested can simply read the abstract/introduction. Many reviews simply quote from the paper anyway. Now that the papers themselves are as easily available as their reviews, it is the good reviews that are of value to the community, but there is no way to guarantee that a review will be good!
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Pete L. ClarkFeb 11 '10 at 6:08

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I do not like that reviews in MathSciNet are locked behind a paywall, despite them being contributed for free by the mathematical community. I never use Zentralblatt, but I tried the webpage just now and it appears to be free. Unfortunately, Zentralblatt appears difficult to use; I was unable to find any actual reviews after a few minutes of frustrated clicking. (I remember the days of using MathReviews on paper in the library, and later via CD's... now that was difficult.)
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William SteinFeb 11 '11 at 6:23

Review whatever you feel comfortable reviewing, whenever you feel ready for it. It's not an obligation, and the AMS Math Reviews and Zentralblatt aren't holding anything over your head. You can refuse to review papers for all kinds of reasons -- you don't have time, you have a conflict of interest, whatever. I wrote my first review 4 years out of grad school.

I would add that the 'time' question is the most important. Also when specifying areas of interest keep them narrow to start with, that way the papers you are sent should be more directly relevant to you thesis work. (Also do discuss the point with your thesis director/supervisor.)
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Tim PorterFeb 8 '10 at 7:25